.
Chapter 4
Exponential, logarithmic and hyperbolic functions
------------------- .
ln y
ax
1
44
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
If you have difficulties taking logarithms of an equation you often may use this trick. Try in an
intermediary step to write both sides as powers of the same base.
Given y e a
Intermediate step
e ln y e a
Result: ln y a
Given: y e a x
…………..=…………..
------------------- 1
45
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
4.1
Powers, Exponential Function
1
First you will study a section in the textbook. For many users this section will be known from school.
But if the material covered is not a mere repetition of well-known facts, carefully take notes and copy
the important rules into your notebook.
Study in the Textbook
4.1.1 Powers; Exponential Function
4.1.2 Laws of Indices or Exponents
Textbook pages 69–71
When done, proceed to
.
------------------- ax
ln y
2
45
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Take logarithms of the following equations
1
y
ex
………….=……………….
y
2 a˜ x
………….=……………….
y 10 ( x 5)
………….=……………….
Choose appropriate bases
------------------- 2
46
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Expand the following expressions
a
4
2
.... ……
b 2
.... ……
------------------- .
ln y
1
x
ldy
a˜x
lg y
x 5
3
46
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
We give some more exercises.
Hint: if exercises seem easy you do not need more of them. If exercises seem difficult you need more
of them
y
b˜ y
e Dx E ………….=……………….
e a˜ x ˜ e c˜ x
………….=……………….
a ˜ y 10 0.1x
y
e ln x ln a ………….=……………….
………….=……………….
------------------- 3
47
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
a4
b
2
a˜a˜a˜a
1
b˜b
3
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Now, let us repeat the vocabulary
The term b m is called ……………….
b is called the ……………………….
m is called …… or ……………..
------------------- .
ln y
ln(by )
Dx E
(a c) x
lgay 0.1x
ln y
4
47
or
ln y
a c x ln b
or
lg y
0.1x lg a
ln x ln a
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
It may be time to have a short break. The reader sketched will have a short break. What is he doing?
Have a guess………….
------------------- 4
48
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
bm is called power
b is named base
m is named exponent or index
Hint: These names should be known by heart
4
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
The definition of powers to a negative exponent is derived by looking at the results of consecutively
dividing a given power by its base.
Express the following power as a fraction.
x 3
.....
------------------- .
Perhaps he recapitulates the new concepts of the preceding section. Perhaps he writes
down the time for the end of his break.
------------------- 5
5
48
49
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
x 3
1
x3
5
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Let us reiterate. The expression
10 x is called …………..
10 is called ……………
x is called …. or ………
------------------- .
4.2.1
Operations with logarithms
6
49
The basic reasoning of operations with logarithms is quite simple. All operations have to be
done with logarithms instead of with the original values.
Thus a product of two values will be the sum of its logarithms.
READ
4.2.2 Operations with logarithms
Textbook pages 76–77
------------------- 6
50
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Power
Base
Exponent or index
6
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Write down the power for:
a) base: x
Exponent: 3 ………….
Power: ………………
b) base A
Exponent x ………….
Power: ……………..
------------------- .
Can you write down
a) ln(a ˜ b)
..................
a
b
.................
b)
ln
7
50
c) lg A ˜ B ................
d)
lg
y
x
.................
------------------- 7
51
Chapter 4 Exponential, logarithmic and hyperbolic functions
a) x3
b) ax
.
7
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
It will be very useful if you understand the origin of the rules. In the following we will deliberately
change the notations. The relationships remain unchanged. Transform the following terms:
ax ˜ay
bm
bn
n m
(x )
Product:
Quotient:
Power:
a
Root:
x
b
.......
.....
....
.....
A t1 ˜ A t2
Sn
Sm
x y
(a )
x
a
y
.....
.....
....
......
------------------- .
a) ln(a ˜ b)
ln a ln b
a
b
ln a ln b
b)
ln
8
51
c) lg A ˜ B log A log B
d)
lg
y
x
lg x lg y
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
In case of difficulties calculate the examples consulting the textbook.
------------------- 8
52
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
A (t1 t2 )
s nm
a x˜ y
Product:
Quotient:
Power:
a x y
b mn
x n˜m
b
y
Root:
xa
ax
8
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
In case of difficulties consult the textbook and solve the exercise again. Do not be disturbed by the
different notations.
In practice notations change in accordance with the given problems.
Now solve
b) (33 ) 0 ...........
a) 27 o .................
c) (2 2 ) 3 ...............
d)
15 .................
------------------- .
In the next examples we deliberately use different notations. The objective is to obtain
a certain familiarity with using different notations.
9
52
lg x ˜ y ..............
ld N 1 ˜ N 2 .............
lg
ln
A˜ B
C
a ˜ E ˜J
G
...............
...........
------------------- 9
53
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
a) 1
b) 1
c) 64
d) 1
9
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Next, compute the values or simplify the expressions respectively.
3 4 ˜ 3 3
a)
b) 10 6 ˜ 10 8 ˜ 10 1
..............
....................
c)
b m
.................
d)
e 1
........................
e)
42
1
...............
------------------- .
lg x lg y
lg xy
ldN 1 ˜ N 2
10
53
ldN 1 ldN 2
lg
AD
c
ln
D ˜ E ˜J
G
lg A lg B lg C
ln D ln E ln J ln G
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Calculate
ln 5 x
.............
lg x 2
.............
1
lg a 2
.............
------------------- 10
54
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
a) 3
10
b) 10
1
bm
c) c
d) 2
e)
1
e
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
If you had difficulties with the previous exercises, solve them consulting the textbook. You should
gain a certain familiarity with these transformations.
If the tasks so far were easy go straight to
If, however, you want to improve your efficiency, you are invited to go to
ln 5 x
x ln 5
lg x 2
2 lg x
.
1
lg a 2
------------------- 17
------------------- 11
54
1
lg a
2
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Transform:
a)
ln 2 x
.................
b)
lg x
................
c)
lg 3 x
...............
d) ld 4 ˜ 16 ...............
------------------- 11
55
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Solve
11
x
a)
A
................
( y 2 )3
c)
e) 10 3 ˜ 10 3 ˜ 10 2
b)
27
1
3
.............
d) (0.1) 0
.................
f)
D 3
.................
..................
.....................
------------------- .
1
lg x
2
a)
ln 2 x
x ln 2
b)
c)
lg 3 x
1
lg x
3
d) ld 4 ˜ 16 ld 4 ld16
lg x
12
55
6
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Can you give the rules
Multiplication: …………
Division: ……………….
Power: …………………
Root: ………………….
------------------- 12
56
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
1
a) A x
b) 3
c) y 6
d) 1
e) 10 6
f)
12
1
D3
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Further exercises wanted
No difficulties so far
.
Multiplication: log AB
A
B
------------------- 13
------------------- 16
log A log B
56
log A log B
Division:
log
Power:
log A m
m log A
Root:
log n A
1
log A
n
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Want to proceed
Want more exercises
13
------------------- 58
------------------- 57
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Solve
a) 2
3
13
0
...........
b) 27
c) e 0
...............
d) 3 1
e) D 3
..............
f)
.................
...............
2
y3
...............
------------------- .
Transform
14
57
a) ln C ˜ D .................
b)
lg y 2
c)
ld 2 ˜ 32
..............
.................
------------------- 14
58
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
a)
1
23
1
8
c) 1
e)
14
b) 1
d)
1
1
3
f) (3 y ) 2
D3
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Transform following this example x n ˜ x m
a) b n b m
..........
b) ( y n ) m
An
Am
...........
d)
c)
n
Cm
x mn
............
..............
------------------- .
a) ln C LnD
15
58
b) 2 lg y
c) 6
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Transform
a)
ld x
...............
b) ln e 2 x ˜ e 5 x ................
c)
lg
1
10 x
..................
------------------- 15
59
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
a) b n m
b) A n m
c) y n˜m
d) C n
15
m
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Last exercise:
1
a) 4 2
b) (30 ) 2
c) 3 4 ˜ 3 3
d) 10 6 ˜ 10 8 ˜ 101
e) e 1
We do not give the answers this time. In case of doubt ask a fellow student or consult the textbook
again.
------------------- .
a)
1
ldx
2
16
59
b) 7x
c) –x
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Calculate by taking logarithms
C
10 3 x 1
x
......................
A
e r ˜t t
......................
16
2 x2
x
......................
------------------- 16
60
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
4.1.1
Exponential function
16
The exponential function is a basic requisite for further studies.
Study in the textbook
4.1.3 Binomial theorem
4.1.4 Exponential function
Pages 71–73
Having done this
.
x
1
lg C 1
3
t
ln A
r
x
2
------------------- 17
60
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
If you encounter difficulties calculating exercises there is a golden rule. Write down the given
exercises on a separate sheet and go back to the textbook.
Try to solve the problem with reference to the textbook and exercise at the same time.
You must not divide your attention by turning over pages many times.
You memorize the operations with logarithms much easier if you understand the relations with the
laws of exponents treated earlier
------------------- 17
61
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
The function y 10 x is called ………………….
17
Which of the following functions increases most rapidly for x o v ?
Insert values for
x = 10, x = 100, x = 1000
y1
x 100
y2
10 x
------------------- .
Can you express the basic reasoning of the operations with logarithms in your own words?
Perhaps you may explain this for a fellow student.
18
61
This is the advantage of working in a team of fellow students. You often need to explain something to
others. It is not enough to understand subject matter. You must be able to express it in your own
words. At the end you will need this competence during your examinations.
------------------- 18
62
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Exponential function
y2
10
18
x
Hint: For x = 1000 we obtain for y1 : (1000)100
y 2 exceeds y1 significantly
10 300 and for y 2 : 101000
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
If we substitute the familiar notations x and y by other notations the mathematical relationships
do not change.
Complex equations often seem difficult if unfamiliar notations are used.
In these cases it often helps to substitute the given notations by x and y and to solve the seemingly
easier equations and then resubstitute.
Sketch the function u
2r
------------------- .
4.2.2
Logarithmic function
READ
19
62
4.2.2 Logarithmic function
Textbook pages 77–78
Having done
------------------- 19
63
Chapter 4 Exponential, logarithmic and hyperbolic functions
19
.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Sketch the exponential function
y 2 at
with a = 2
------------------- .
At what point intersect all logarithmic functions?
x
..................
y
.................
20
63
Has the logarithmic function a pole?
yes
no
Has the logarithmic function an asymptote
yes
no
------------------- 20
64
Chapter 4 Exponential, logarithmic and hyperbolic functions
20
.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Solve the following task.
In case of difficulties consult the textbook.
The plot represents the general exponential
function
t
y A ˜ e tn
Determine A and t h (t h
Use the given points.
y = ………………..
half life value)
------------------- .
Intersection of all logarithmic functions at x = 1
y=0
21
64
Pole at x = 0
No asymptote
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Sketch the graph of
y
y
lg x
ln x
------------------- 21
65
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
y 10 ˜ 2
t
2
21
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
All correct
Explanation wanted
------------------- 24
------------------- 22
65
.
------------------- 22
66
Chapter 4 Exponential, logarithmic and hyperbolic functions
In the textbook this exponential function is
explained. t is the time. The initial value for t = o is
A = ……………..
From the plot you may read that the curve decreases
to half of its initial value at t h ...........
.
22
Hint:
t
At t = 0
the term A ˜ e th
Thus, the function is
A ˜ e0
.........
y = ………………
------------------- .
The logarithmic function is the inverse of the exponential function.
Is the exponential function the inverse of the logarithmic function?
yes
no
23
23
66
------------------- 68
------------------- 67
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
A 10
th
23
2
y 10 ˜ 2
t
2
Hint: The values have been taken from the graph.
Since A = 10 and t h
2 we insert into y
A˜e
t
th
and obtain y 10 ˜ 2
t
2
------------------- .
Sorry you are wrong. We obtain the inverse of a function by reflecting it in the line y = x
which bisects the first quadrant and vive versa. Go back to section 3.4 “inverse functions”
and read it again. The relation is symmetrical.
------------------- 24
24
67
68
Chapter 4 Exponential, logarithmic and hyperbolic functions
Sketch the exponential function given below
.
F
e 0.5t
24
e = 2.72
In case you are not familiar with this notation you
may substitute F by y and t by x
Perhaps the equation will be more familiar to you
------------------- .
Yes, you are right. The exponential function is the inverse function of the
logarithmic function.
We remember that we get the inverse function by reflecting the original function
in the line y x which bisects the first quadrant.
25
68
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Not every function has an inverse function.
If possible give the inverse function for
1
..............
1
.............
y1
32 x
y1
y2
4x 2
y2
Hint:
Remember the definition of a function.
------------------- 25
69
Chapter 4 Exponential, logarithmic and hyperbolic functions
25
.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
We suggest a break. You may work with fairly good concentration for 20 minutes up to 60 minutes.
There are great differences in optimal individual working periods.
If you are interested in your work these periods may be longer.
Find out how long you may work with concentration. It is important to divide your tasks and to have
a short break - and to end the short break in due time..
------------------- .
y1
1
1
x
log 3
2
26
69
y 2 has no inverse function since the expression y 2
1
r
1
x does not represent a function because
2
it is ambiguous.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
On the other side the sine curve is plotted.
You may reflect it in the line y x
which bisects the first quadrant.
Sketch the reflected curve. Is the reflected
curve a function?
------------------- 26
70
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Short breaks have a duration of 5–15 minutes. With longer breaks the difficulties of warm up
rise again.
It is of some importance what you do during the break. Mark appropriate activities
26
Water your flowers, have a cup of tea, prepare a coffee, do some physical exercise
solve mathematical problems, read a different chapter in your textbook.
------------------- .
The reflected curve is ambiguous. It is a
relation not a function. We can get a
function if we restrict the domain and the
codomain.
27
70
S
S
dyd
2
2
You know this function already:
y arcsin x and you know its meaning.
Complete:
y is the angle …………………………..
………………………….
Sketch in the plot above with a heavy line the function for the restricted domain and codomain.
------------------- 27
71
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Appropriate activities are to water flowers, prepare tea or coffee, do some physical exercise.
27
Learning and memorizing a certain matter will be prevented if you concentrate during your break on
a similar matter. Solving other mathematical problems is very similar to your learning. It has a
negative effect on your learning. This phenomenon is well known in psychology and is named
interference.
Another suggestion. Write down on a separate sheet of paper the planned end of your break.
Now enjoy your break.
------------------- .
y is the angle whose sine is x.
28
71
------------------- 28
72
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Short break
28
Having ended your break go to
.
4.3
------------------- Hyperbolic functions and inverse Hyperbolic Functions
29
72
Since these functions are used in advanced mathematics chapters you may skip this section
for the time being. Especially if you found it difficult to master the preceding two introductory
chapters.
In this case you may return to this section later.
But if all of the preceding material was known you should study this section now.
I want to skip section 4.3
------------------- 88
I want to study the section on hyperbolic functions
READ
4.3. Hyperbolic functions and inverse hyperbolic functions
Textbook pages 78–82
Having done
------------------- 29
73
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Before continuing compare the actual time with the time you planned. There may be a
difference. Breaks tend to increase. This does not matter. But in the long run these differences
between planned lengths of breaks and the real lengths should not increase too much.
------------------- .
Give the definition of
sinh x
29
30
73
....................
------------------- 30
74
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
4.2
Logarithm, logarithmic function
30
Logarithms are quite difficult when learned for the first time; however if you are familiar
with this matter you will proceed quickly.
READ
4.2.1 Logarithm
Textbook pages 74–75
Then go to
------------------- .
sinh x
e x ex
2
31
74
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Sketch y1 sinh 2 x with a dashed line
x
and y 2 sinh with a dotted line
2
------------------- 31
75
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Taking logarithms is a new operation. To take logarithms is to solve the equation y
for x. This means given: ………….. and ……………
wanted ……….…
ax
------------------- 31
32
75
.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Give the definition of cosh x
................
------------------- 32
76
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
y ax
given: a and y
32
wanted: x
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
In another notation the task of taking logarithms can be expressed as well:
The equation a y x is to be solved for y.
Up to now we are unable to find a solution.
Thus, we must create a new operation. In mathematics this new operation is called “Taking
logarithms.”
------------------- .
cosh x
e x ex
2
33
76
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
cosh 2 x with a dashed line
x
cosh with a dotted line
2
Sketch y1
and y 2
------------------- 33
77
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Let us take the equation a y x
In the textbook you found the following definition:
33
The logarithm of a given number x to a base a is the exponent of the power to which this base
must be raised to equal this number x.
For this exponent we use the symbol
log a x
In other words: The term “ log a x ” is a power or an exponent.
a (log a x)
................
------------------- 34
77
.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Calculate cosh 2 x sinh 2 x 2
2
.......................
------------------- 34
78
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
a (log a x)
x
34
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
You must memorize:
For a given base
the logarithm of a certain number is the exponent to equal this number.
The definition and the meaning of logarithms are important and you must be familiar with them. The
logarithms for base 10 are called ………… and abbreviated ……………….
------------------- .
cosh 2 x 2 sinh 2 x 2
1
35
78
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Give the definition of tanh x . Try first to answer without consulting the textbook
tanh x
............. ................... ...............................
------------------- 35
79
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
common logarithm abbreviation: lg or log
35
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Let us regard common logarithms.
Calculate:
10 log 5
.................
10 lg 20
.................
10 log 3.14
................
------------------- .
tanh x
sinh x
cosh x
e x ex
e x e x
1 e 2 x
1 e 2 x
36
79
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Try to sketch
y1
y2
tanh 2 x with a dashed line and
x
with a dotted line
tanh
2
------------------- 36
80
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
10 log 5
10 lg 20
10 log 3.14
5
20
3.14
36
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Logarithms with base 2 are called dyadic logarithms or logarithms to the base of 2 and abbreviated
…………….
Calculate
2 ld 4
............
2 ld 100
............
2 ldb
............
------------------- 37
80
.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Try to sketch y1 coth x with a dashed line
x
and y 2 coth with a dotted line
2
------------------- 37
81
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Logarithms to the base of 2 are abbreviated ld
2 ld 4 4
2 ld 100
2 ldb
37
100
b
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Logarithms with base e are called ………….. and abbreviated ……………….
Calculate:
e lm 6
..............
e lma
.............
e lm10
.............
------------------- 38
81
.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
In the textbook we give the definition of the inverse hyperbolic functions.
If you wish to derive one of them
If you want to skip derivation of sinh x 1
38
------------------- 82
------------------- 87
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
Natural Logarithms
ln 6
6
e ln a
a
e ln 10
10
e
ln
38
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
With logarithms the base has always to be defined. Give the names of the logarithms with the
following base.
base 2: ………………..
base e: ………………..
base 10: ………………
------------------- .
Given the hyperbolic sine: f ( x)
We may write it y
sinh x
1 x
e e x 2
39
82
1 x
e e x 2
To obtain the inverse function we change y and x
x = ……………………
------------------- 39
83
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
base 2: dyadic logarithms or logarithms to the base of 2
base e: natural logarithms
base 10: common logarithms
39
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
If you want to take logarithms you have three options
1. You use your pocket calculator or your computer. This is convenient and precise.
2. You use a plot of the logarithmic function. This is convenient but not precise.
3. You use a table. This is precise but inconvenient.
For some values you can calculate the logarithms without help
ld 2
..................
ln e x
............... ..
log100 .............. …
------------------- .
x
2x
1 y
e e y 2
e
y
40
83
e y Multiplying by e y gives
2x ˜ e y
.................
------------------- 40
84
Chapter 4 Exponential, logarithmic and hyperbolic functions
.
ld 2 1
since 21
2
ex
ln e x
x
since e x
log 100
3
since 10 3
40
100
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An operation used later on is to take logarithms of equations. In this case the operation is to be
applied to both sides of the equation.
By this operation equations sometimes simplify.
Example: given e y e ax
The base is (and must be) the same for both sides. In this case the exponents on both sides must be
equal:
y ax
Thus, we have just taken the logarithm of the equation since ln e y
Take the logarithms
e a e bc
y
ln e ax
ax
…………..=…………….
------------------- .
2 xe y
e
2y
1
Substituting e y
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84
a
gives
2 xa
...............
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Chapter 4 Exponential, logarithmic and hyperbolic functions
.
a=b+c
41
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Take the logarithm of the equation
10 y 10 bx
Using common logarithms we get
lg 10 y lg 10 bx
Thus y bx
Take logarithms of the following equations
2y
2 cx
y =…………………
ea
e Z ( t t0 )
a =………………….
------------------- .
2 xa
a2 1
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Thus, we get a quadratic function for a.
We already know to solve it:
a
x x2 1
Since a
ey
e y we obtain
x x2 1
To obtain the inverse function y we take the logarithm and obtain y =……………..
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Chapter 4 Exponential, logarithmic and hyperbolic functions
.
y
cx
a
Z (t t 0 )
42
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To take logarithms of an equation means to regard exponents if the bases are equal.
Example: 2 7 2 x 1
ld 2 7 ld 2 x 1
We take the logarithms:
7 x 1
x 6
10 ( 2 y 1)
y
Calculate
10 x 3
............
------------------- .
y
ln x x 2 1
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Well done.
If you worked through this rather rough matter you will use hyperbolic functions later on in advanced
mathematics.
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Chapter 4 Exponential, logarithmic and hyperbolic functions
.
y
1
( x 4)
2
43
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In the foregoing examples we had on both sides of the equation exponents of the same base. This is
not always the case.
Example y e ax . Can you take the logarithm? You may think this is impossible. But in this case a
trick helps.
Write down y as a power of e.
y e ln y
Thus, we have on both sides the same base
e ln y e ax
Now you can take the logarithm of the equation:
…………..=…………….
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Please continue on page 1
(bottom half)
87
.
You have reached the end of chapter 4
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